# Summary

My advisor developed a framework for solving a large class of problems of particular interest to the engineering disciplines. Starting as a graduate research assistant under his direction in the summer of 2015, I began to learn about this new method and explore implications of how measurements of physical systems are taken on our ability to accurately describe parameter uncertainty sets.

You can read more about it below, but overall, my focus on *skewness* ended up playing a less important role when we developed a sample-based approach to solving the same problems.
When I performed some of my Master’s experiments with this new approach, almost all the effects on error disappeared.
I later turned my attention to extending this sample-based approach to problems concerning parameter identification under uncertainty.

# Abstract

Over the past several years, a measure-theoretic framework for formulating and solving stochastic inverse problems has been developed and analyzed. In this framework, we begin with a model, model inputs (called parameters), and a quantity of interest map that defines the types of model outputs used to formulate the stochastic inverse problem.

The solution to the stochastic inverse problem is a (non- parametric) probability measure on the space of parameters. A non-intrusive sample based algorithm has also been developed and analyzed to approximate this probability measure. More recent work has shown that a particular geometric property (the skewness) of the quantity of interest map is correlated to the accuracy in the finite sample approximation of the probability measure.

In this work, we provide a more in-depth numerical investigation into how skewness impacts the accuracy and convergence rates of such approximate probability measures. We provide an algorithmic procedure for computing a metric on a space of probability measures, and we demonstrate that while convergence rates are unaffected by the choice of map, the absolute error is directly proportional to the skewness of the map.